1. Field of the Invention
The present invention relates generally to circuitry for use in the equalizer devices of digital communications systems, and more specifically to an adaptive leaky digital integrator circuit having particular application in the instability compensation network of an equalizer circuit of the type used in modem devices.
2. Description of the Prior Art
It is well known in communication technology that during transmission of digital data to a remote receiver, the transmitted signal often suffers amplitude and phase shift distortions that in many cases make the signal very difficult to decode after it has been received. In order to compensate for such nonlinearities caused by the transmission media, and particularly in systems in which the media is a telephone channel, it has become common practice to utilize equalizing circuitry in the receiver front end.
A typical band-limited telephone channel can be characterized by its transfer function H(f) which is the Fourier transform of the channel impulse response h(t). In FIG. 1 of the drawing, a typical impulse response (in discrete time) is illustrated which shows the substantial inter-symbol interference which can occur if an attempt is made to signal at a rate which is too high. In discrete time a telephone channel can be modeled by a feed-forward or transversal filter such as that shown in FIG. 2 of the drawing. Note that the tap values correspond to the impulse response at the corresponding delay values.
The harmful effects of inter-symbol interference can be removed however by processing the received signal using an adaptive equalizer having a transfer function H.sup.-1 (f). Thus, the series combination of the transfer functions H(f) and H.sup.-1 (f) is equivalent to multiplication by 1. That is, in the noise free case, the output of the equalizer is the same as the channel input, i.e., the transmitter output. FIG. 3 illustrates an overall system block diagram in which x(t) is the received signal and y(t) is the processed signal.
It may be shown that any equalizer transfer function can be realized by the linear transversal filter schematically illustrated in FIG. 4. In such circuit the equalizer output is given by ##EQU1## where a.sub.i represents the tap coefficients of the transversal filter and x(t) is the channel output, i.e., the equalizer input. Note that it may be more computationally efficient to use feedback or partial feedback (and partial transversal) equalization. However, for the present explanatory purposes, it suffices to consider a transversal equalizer.
Designing an equalizer to produce the transfer function H.sup.-1 (f) is an optimum strategy provided one knows H(f) and can compute H.sup.-1 (f). Unfortunately, in practice H(f) is seldom known and in effect must be learned. However, rather than learning H(f) and computing H.sup.-1 (f), it has been found that the least mean square (LMS) algorithm can be applied directly to computing the tap coefficients. The LMS algorithm states that at any time "t" the i.sup.th tap coefficient a.sub.i (t) can be related to its time predecessor a.sub.i (t-1) by the equation EQU a.sub.i (t)=a.sub.i (t-1)+.DELTA.a.sub.i (t)
where .DELTA.a.sub.i (t) is proportional to ##EQU2## In the above expression e(n) is the equalization signal which is given by e(n)=m(n)-y(n) and m(n) is the detector output signal. Thus, a simple cross correlation can be used to update the tap coefficients in such a fashion that the equalizer produces the inverse of the channel transfer function.
However, adaptive equalizers used in telephone line data modems can have stability problems in that for channels which are narrowband for the signaling rate, the equalizer tap coefficients may drift away from the optimum values. Note that if in the above equation a.sub.i (t-1) has an error, that error is also present in a.sub.i (t), a.sub.i (t-1), etc. That is, there is no mechanism for causing the error to be "forgotten." A reference to instability correction by G. Underboeck entitled "Fractional Tap-Spacing Equalizer and Cosequences for Clock Recovery in Data Modems," may be found in IEEE Trans Communications, Vo. COM-24, No. 8, pp. 856-864 (August 1976).
One method of combatting this problem is to replace the ideal integrator normally used in implementing the tap coefficient update algorithm stated above. More specifically, the update equation is modified by including a decay factor .beta. in the time predecessor component; that is, by making EQU a.sub.i (t)=.beta.a.sub.i (t-1)+.DELTA.a.sub.i (t).
For reasonable performance, the parameter .beta. should be quite close to but less than 1; e.g., 0.999.
In FIG. 5 of the drawing, a prior art equalizer and its adaptive tap coefficient network is shown at 10 which includes a shift register 11, a correlator 12, commutator switches 13 and 14, a multiplier 15, an accumulator 16 and a plurality of leaky integrators I.sub.1 -I.sub.n provided in the tap coefficient update circuits. Note that as illustrated within the dashed lines, the integrator I.sub.1, for example, includes an adder 17, a shift register 18, a multiplier 19 and a ROM 20. An update increment .DELTA.a.sub.i (t) is input to each integrator I from the correlator 12 and is added in adder 17 to the delayed coefficient a.sub.i (t-1) after it has been multiplied by a decay factor .beta. which is extracted from the ROM 20.
The difficulty with this theoretically sound approach for implementing a leaky integrator is that it requires one multiplication per tap coefficient per input sample processed by the equalizer. For many practical systems, this requires an excessive number of multiplications per second and thus an inordinant increase in the complexity and cost of the system.
Prior art references discussing leaky integrators are Delta Modulator Systems by R. Steele, Halsted Press, John Wiley & Sons (ISBN 0-470-82104-3), p. 304; Linear Systems in Communication and Control by D. K. Frederick and A. B. Carlson, John Wiley & Sons (ISBN 0-471-27721-5), p. 449; and Digital Processing of Signals by Gold and Rader, McGraw Hill, Library of Congress No. 69-13606, pp. 23-24.